scholarly journals The Galois algebras and the Azumay Galois extensions

2002 ◽  
Vol 31 (1) ◽  
pp. 37-42
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Galois Group ◽  
Commutative Ring ◽  
Galois Extension ◽  
Galois Extensions

LetBbe a Galois algebra over a commutative ringRwith Galois groupG,Cthe center ofB,K={g∈G|g(c)=c for all c∈C},Jg{b∈B|bx=g(x)b for all x∈B}for eachg∈K, andBK=(⊕∑g∈K Jg). ThenBKis a central weakly Galois algebra with Galois group induced byK. Moreover, an Azumaya Galois extensionBwith Galois groupKis characterized by usingBK.

scholarly journals On the Quadratic Extensions and the Extended Witt Ring of a Commutative Ring

1973 ◽  
Vol 49 ◽  
pp. 127-141 ◽  
Author(s):  
Teruo Kanzaki
Keyword(s):  
Galois Group ◽  
Commutative Ring ◽  
Galois Extension ◽  
Identity Element ◽  
Quadratic Extension ◽  
Crossed Product ◽  
Witt Ring ◽  
Division Rings ◽  
Quadratic Extensions ◽  
The Common

Let B be a ring and A a subring of B with the common identity element 1. If the residue A-module B/A is inversible as an A-A- bimodule, i.e. B/A ⊗A HomA(B/A, A) ≈ HomA(B/A, A) ⊗A B/A ≈ A, then B is called a quadratic extension of A. In the case where B and A are division rings, this definition coincides with in P. M. Cohn [2]. We can see easily that if B is a Galois extension of A with the Galois group G of order 2, in the sense of [3], and if is a quadratic extension of A. A generalized crossed product Δ(f, A, Φ, G) of a ring A and a group G of order 2, in [4], is also a quadratic extension of A.

scholarly journals On central commutator Galois extensions of rings

2000 ◽  
Vol 24 (5) ◽  
pp. 289-294
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Automorphism Group ◽  
Galois Group ◽  
Group Ring ◽  
Galois Extension ◽  
Structure Theorem ◽  
Projective Group ◽  
Galois Extensions ◽  
Separable Extensions

LetBbe a ring with1,Ga finite automorphism group ofBof ordernfor some integern,BGthe set of elements inBfixed under each element inG, andΔ=VB(BG)the commutator subring ofBGinB. Then the type of central commutator Galois extensions is studied. This type includes the types of Azumaya Galois extensions and GaloisH-separable extensions. Several characterizations of a central commutator Galois extension are given. Moreover, it is shown that whenGis inner,Bis a central commutator Galois extension ofBGif and only ifBis anH-separable projective group ringBGGf. This generalizes the structure theorem for central Galois algebras with an inner Galois group proved by DeMeyer.

scholarly journals Mahler measure on Galois extensions

2018 ◽  
Vol 14 (06) ◽  
pp. 1605-1617 ◽  
Author(s):  
Francesco Amoroso
Keyword(s):  
Symmetric Group ◽  
Galois Group ◽  
Galois Extension ◽  
Mahler Measure ◽  
Algebraic Numbers ◽  
Galois Extensions

We study the Mahler measure of generators of a Galois extension with Galois group the full symmetric group. We prove that two classical constructions of generators give always algebraic numbers of big height. These results answer a question of Smyth and provide some evidence to a conjecture which asserts that the height of such a generator grows to infinity with the degree of the extension.

scholarly journals The Galois extensions induced by idempotents in a Galois algebra

2002 ◽  
Vol 29 (7) ◽  
pp. 375-380
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Galois Group ◽  
Galois Extension ◽  
Central Idempotent ◽  
Galois Extensions ◽  
Direct Sum ◽  
Equivalence Condition ◽  
Minimal Idempotent

LetBbe a Galois algebra with Galois groupG,Jg={b∈B|bx=g(x)b   for all   x∈B}for eachg∈G,egthe central idempotent such thatBJg=Beg, andeK=∑g∈K,eg≠1egfor a subgroupKofG. ThenBeKis a Galois extension with the Galois groupG(eK)(={g∈G|g(eK)=eK})containingKand the normalizerN(K)ofKinG. An equivalence condition is also given forG(eK)=N(K), andBeGis shown to be a direct sum of allBeigenerated by a minimal idempotentei. Moreover, a characterization for a Galois extensionBis shown in terms of the Galois extensionBeGandB(1−eG).

scholarly journals On weak center Galois extensions of rings

2001 ◽  
Vol 25 (7) ◽  
pp. 489-495
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Automorphism Group ◽  
Galois Group ◽  
Galois Extension ◽  
Galois Extensions ◽  
Weak Center

LetBbe a ring with 1,Cthe center ofB,Ga finite automorphism group ofB, andBGthe set of elements inBfixed under each element inG. Then, the notion of a center Galois extension ofBGwith Galois groupG(i.e.,Cis a Galois algebra overCGwith Galois groupG|C≅G) is generalized to a weak center Galois extension with groupG, whereBis called a weak center Galois extension with groupGifBIi=Beifor some idempotent inCandIi={c−gi(c)|c∈C}for eachgi≠1inG. It is shown thatBis a weak center Galois extension with groupGif and only if for eachgi≠1inGthere exists an idempotenteiinCand{bkei∈Bei;ckei∈Cei,k=1,2,...,m}such that∑k=1mbkeigi(ckei)=δ1,gieiandgirestricted toC(1−ei)is an identity, and a structure of a weak center Galois extension with groupGis also given.

Galois Extensions as Modules over the Group Ring

1970 ◽  
Vol 22 (2) ◽  
pp. 242-248 ◽  
Author(s):  
Gerald Garfinkel ◽  
Morris Orzech
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Galois Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Finite Abelian Group ◽  
Galois Extensions ◽  
Normal Bases ◽  
Direct Sum ◽  
Definition Of

Suppose that R is a commutative ring and G is a finite abelian group. In § 2 we review the definition of E(R, G) (T(R, G)), the group of all (commutative) Galois extensions S of R with Galois group G. We discuss the properties of these groups as functors of G and give an example which exhibits some of the pathological properties of the functor E(R, – ). In § 3 we display a homomorphism from E(R, G) to Pic (R(G)); we use this homomorphism to prove that if S is commutative, G has exponent m, and R(G) has Serre dimension 0 or 1, then a direct sum of m copies of S is isomorphic as a G-module to a direct sum of m copies of R(G). (This result is related to [5, Theorem 4.2], where it is shown that if S is a free R-module and G is any finite group with n elements, then Sn is isomorphic to R(G)n as G-modules.) We also give some examples of Galois extensions without normal bases.

scholarly journals Integral Normal Bases in Galois Extensions of Local Fields

1970 ◽  
Vol 39 ◽  
pp. 141-148 ◽  
Author(s):  
S. Ullom
Keyword(s):  
Prime Ideal ◽  
Galois Group ◽  
Galois Extension ◽  
Residue Field ◽  
Local Fields ◽  
Galois Extensions ◽  
Normal Bases ◽  
Ring Of Integers

Throughout this paper F denotes a field complete with respect to a discrete valuation, kF the residue field of F, K/F a finite Galois extension with Galois group G = G(K/F). The ring of integers 0K of K contains the (unique) prime ideal ; the collection of ideals n for all integers n are ambiguous ideals i.e. G-modules.

scholarly journals On Azumaya Galois extensions and skew group rings

1999 ◽  
Vol 22 (1) ◽  
pp. 91-95
Author(s):  
George Szeto
Keyword(s):  
Galois Group ◽  
Group Ring ◽  
Galois Extension ◽  
Group Rings ◽  
Galois Extensions ◽  
Skew Group Rings ◽  
Skew Group Ring

Two characterizations of an Azumaya Galois extension of a ring are given in terms of the Azumaya skew group ring of the Galois group over the extension and a Galois extension of a ring with a special Galois system is determined by the trace of the Galois group.

scholarly journals Cusp forms of weight one, quartic reciprocity and elliptic curves

1985 ◽  
Vol 98 ◽  
pp. 117-137 ◽  
Author(s):  
Noburo Ishii
Keyword(s):  
Positive Integer ◽  
Elliptic Curves ◽  
Galois Group ◽  
Cusp Form ◽  
Rational Number ◽  
Galois Extension ◽  
Dihedral Group ◽  
Cusp Forms ◽  
Two Dimensional ◽  
Weight One

Let m be a non-square positive integer. Let K be the Galois extension over the rational number field Q generated by and . Then its Galois group over Q is the dihedral group D4 of order 8 and has the unique two-dimensional irreducible complex representation ψ. In view of the theory of Hecke-Weil-Langlands, we know that ψ defines a cusp form of weight one (cf. Serre [6]).

Cleft extensions and Galois extensions for Hom-associative algebras

2016 ◽  
Vol 27 (03) ◽  
pp. 1650025 ◽  
Author(s):  
J. N. Alonso Álvarez ◽  
J. M. Fernández Vilaboa ◽  
R. González Rodríguez
Keyword(s):  
Associative Algebra ◽  
Galois Extension ◽  
Classical Case ◽  
Normal Basis ◽  
Associative Algebras ◽  
Galois Extensions ◽  
Cleft Extension ◽  
Cleft Extensions

In this paper, we consider Hom-(co)modules associated to a Hom-(co)associative algebra and define the notion of Hom-triple. We introduce the definitions of cleft extension and Galois extension with normal basis in this setting and we show that, as in the classical case, these notions are equivalent in the Hom setting.

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